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Mirrors > Home > HSE Home > Th. List > ho0val | Structured version Visualization version GIF version |
Description: Value of the zero Hilbert space operator (null projector). Remark in [Beran] p. 111. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ho0val | ⊢ (𝐴 ∈ ℋ → ( 0hop ‘𝐴) = 0ℎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | choc1 31252 | . . . . . 6 ⊢ (⊥‘ ℋ) = 0ℋ | |
2 | 1 | fveq2i 6903 | . . . . 5 ⊢ (projℎ‘(⊥‘ ℋ)) = (projℎ‘0ℋ) |
3 | df-h0op 31673 | . . . . 5 ⊢ 0hop = (projℎ‘0ℋ) | |
4 | 2, 3 | eqtr4i 2756 | . . . 4 ⊢ (projℎ‘(⊥‘ ℋ)) = 0hop |
5 | 4 | fveq1i 6901 | . . 3 ⊢ ((projℎ‘(⊥‘ ℋ))‘𝐴) = ( 0hop ‘𝐴) |
6 | helch 31168 | . . . 4 ⊢ ℋ ∈ Cℋ | |
7 | pjo 31596 | . . . 4 ⊢ (( ℋ ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ((projℎ‘(⊥‘ ℋ))‘𝐴) = (((projℎ‘ ℋ)‘𝐴) −ℎ ((projℎ‘ ℋ)‘𝐴))) | |
8 | 6, 7 | mpan 688 | . . 3 ⊢ (𝐴 ∈ ℋ → ((projℎ‘(⊥‘ ℋ))‘𝐴) = (((projℎ‘ ℋ)‘𝐴) −ℎ ((projℎ‘ ℋ)‘𝐴))) |
9 | 5, 8 | eqtr3id 2779 | . 2 ⊢ (𝐴 ∈ ℋ → ( 0hop ‘𝐴) = (((projℎ‘ ℋ)‘𝐴) −ℎ ((projℎ‘ ℋ)‘𝐴))) |
10 | 6 | pjhcli 31343 | . . 3 ⊢ (𝐴 ∈ ℋ → ((projℎ‘ ℋ)‘𝐴) ∈ ℋ) |
11 | hvsubid 30951 | . . 3 ⊢ (((projℎ‘ ℋ)‘𝐴) ∈ ℋ → (((projℎ‘ ℋ)‘𝐴) −ℎ ((projℎ‘ ℋ)‘𝐴)) = 0ℎ) | |
12 | 10, 11 | syl 17 | . 2 ⊢ (𝐴 ∈ ℋ → (((projℎ‘ ℋ)‘𝐴) −ℎ ((projℎ‘ ℋ)‘𝐴)) = 0ℎ) |
13 | 9, 12 | eqtrd 2765 | 1 ⊢ (𝐴 ∈ ℋ → ( 0hop ‘𝐴) = 0ℎ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6553 (class class class)co 7423 ℋchba 30844 0ℎc0v 30849 −ℎ cmv 30850 Cℋ cch 30854 ⊥cort 30855 0ℋc0h 30860 projℎcpjh 30862 0hop ch0o 30868 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-inf2 9680 ax-cc 10474 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 ax-pre-sup 11232 ax-addf 11233 ax-mulf 11234 ax-hilex 30924 ax-hfvadd 30925 ax-hvcom 30926 ax-hvass 30927 ax-hv0cl 30928 ax-hvaddid 30929 ax-hfvmul 30930 ax-hvmulid 30931 ax-hvmulass 30932 ax-hvdistr1 30933 ax-hvdistr2 30934 ax-hvmul0 30935 ax-hfi 31004 ax-his1 31007 ax-his2 31008 ax-his3 31009 ax-his4 31010 ax-hcompl 31127 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-se 5637 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7689 df-om 7876 df-1st 8002 df-2nd 8003 df-supp 8174 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-oadd 8499 df-omul 8500 df-er 8733 df-map 8856 df-pm 8857 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9402 df-fi 9450 df-sup 9481 df-inf 9482 df-oi 9549 df-card 9978 df-acn 9981 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-div 11918 df-nn 12260 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-7 12327 df-8 12328 df-9 12329 df-n0 12520 df-z 12606 df-dec 12725 df-uz 12870 df-q 12980 df-rp 13024 df-xneg 13141 df-xadd 13142 df-xmul 13143 df-ioo 13377 df-ico 13379 df-icc 13380 df-fz 13534 df-fzo 13677 df-fl 13807 df-seq 14017 df-exp 14077 df-hash 14343 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-clim 15485 df-rlim 15486 df-sum 15686 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-starv 17276 df-sca 17277 df-vsca 17278 df-ip 17279 df-tset 17280 df-ple 17281 df-ds 17283 df-unif 17284 df-hom 17285 df-cco 17286 df-rest 17432 df-topn 17433 df-0g 17451 df-gsum 17452 df-topgen 17453 df-pt 17454 df-prds 17457 df-xrs 17512 df-qtop 17517 df-imas 17518 df-xps 17520 df-mre 17594 df-mrc 17595 df-acs 17597 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-submnd 18769 df-mulg 19057 df-cntz 19306 df-cmn 19775 df-psmet 21327 df-xmet 21328 df-met 21329 df-bl 21330 df-mopn 21331 df-fbas 21332 df-fg 21333 df-cnfld 21336 df-top 22879 df-topon 22896 df-topsp 22918 df-bases 22932 df-cld 23006 df-ntr 23007 df-cls 23008 df-nei 23085 df-cn 23214 df-cnp 23215 df-lm 23216 df-haus 23302 df-tx 23549 df-hmeo 23742 df-fil 23833 df-fm 23925 df-flim 23926 df-flf 23927 df-xms 24309 df-ms 24310 df-tms 24311 df-cfil 25266 df-cau 25267 df-cmet 25268 df-grpo 30418 df-gid 30419 df-ginv 30420 df-gdiv 30421 df-ablo 30470 df-vc 30484 df-nv 30517 df-va 30520 df-ba 30521 df-sm 30522 df-0v 30523 df-vs 30524 df-nmcv 30525 df-ims 30526 df-dip 30626 df-ssp 30647 df-ph 30738 df-cbn 30788 df-hnorm 30893 df-hba 30894 df-hvsub 30896 df-hlim 30897 df-hcau 30898 df-sh 31132 df-ch 31146 df-oc 31177 df-ch0 31178 df-shs 31233 df-pjh 31320 df-h0op 31673 |
This theorem is referenced by: df0op2 31677 hoaddridi 31711 ho0coi 31713 0cnop 31904 0hmop 31908 nmop0 31911 adj0 31919 nmlnop0iALT 31920 lnopco0i 31929 lnopeq0i 31932 nmopcoi 32020 leop3 32050 leoprf2 32052 leoprf 32053 idleop 32056 pjorthcoi 32094 |
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